Development of atomic theory

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2 Development of atomic theory The chapter presents the fundamentals needed to explain and atomic & molecular structures in qualitative or semiquantitative terms.

3 Li B B C N O F Ne Sc Ti V Cr Mn Fe Co Ni Cu Zn

5 Thomson s experiment

6 This deflection depends on: a. the strength of the deflecting magnetic or electric field b. the size of the negative charge on the electron c. the mass of the electron * Charge-to-mass ratio, e/m of the electron = C/g.

7 Millikan s experiment

8 Millikan s experiment Determined that the charge on a drop of oil was whole-number multiple of e (e = C). Knowing the values for e/m and e for an electron, m can be calculated (m = g).

9 Rutherford s experiment

12 In 1885, Balmer showed the energies of visible light emitted by the H atom are given by the equation. E=h

14 The Balmer eq. was later made more general, as spectral lines in the ultraviolet & infrared regions of the spectrum were discovered, by replacing 2 2 by n l2, with the condition that n l < n h.

16 The theory (Bohr s quantum theory of atom) assumed that negative electrons in atoms move in stable circular orbits around the positive nucleus with no absorption or emission of energy (however, electrons may absorb/emit light of specific energies). E=h

17 Classical physics : a particle in motion tends to move in a straight line or in a circle by application of a force toward the center of the circle. Since an electron revolving around the nucleus constantly changes its direction, it is constantly accelerating. Therefore, the electron should emit light & lose energy, & must be drawn into the nucleus. This conclusion does not correlate with the existence of stable atoms.

18 When applied to hydrogen, Bohr s theory worked well; when atoms with more electrons were considered, the theory failed (i.e., elliptical rather than circular orbits).

22 410.1 nm

23 Particles massive enough to be visible have very short wavelengths, too small to be measured. Electrons, on the other hand, have observable wave properties because of their very small mass.

24 Hsisenberg s uncertainty principle states that there is a relationship between the inherent uncertainty in the location & momentum of an electron. The x component of this uncertainty is described as

25 The energy of spectral lines can be measured with great precision, in turn allowing precise determination of the energy of electrons in atoms. This precision in energy also implies precision in momentum ( p x is small); therefore according to Heisenberg, there is a large uncertainty in the location of the electron ( x is large).

26 For an electron, ~ 1s orbital of H atom (240 pm)

27 For a car,

28 This means that we cannot treat electrons as simple particles with their motions described precisely, but we must instead consider the wave properties of electrons, characterized by a degree on uncertainty in their location.

29 We must change : Orbits (Bohr) Orbitals (regions that describe the probable location of electrons) The probability of finding the electron at a particular point in space, also called the electron density ( ), can be calculated at lease in principle.

30 2-2 The SchrÖdinger Equation The SchrÖdinger equation describes the wave properties of an electron in terms of its position, mass, total energy, and potential energy. The equation is based on the wave function,, which describes an electron wave in space; in other words, it describes an atomic orbital.

32 In the form used for calculating energy levels, the Hamiltonian is

35 The potential energy, V, is a result of electrostatic attraction between the electron & the nucleus. Attractive forces, like those between a positive nucleus & a negative electron, are defined by convention to have a negative potential energy.

36 (1) Because every atomic orbital is described by a unique, there is no limit to the # of solutions of the SchrÖdinger equation for an atom. (2) Each describes the wave properties of a given electron in a particular orbital. (3) The probability of finding an electron at a given point in space is proportional to 2.

37 A number of conditions are required for a physically realistic solution for :

38 For complex and real numbers

39 2.2.1 The particle in a box H = E A general solution = Asin rx Apply x=0 & a, V=0.

40 The Particle in a box as a model - A particle (m) is confined at 0 < x <L (V=0), but V= at x=0 & L. H =E (E=kinetic energy), H = -ħ 2 /2m(d 2 /dx 2 ) -ħ 2 /2m(d 2 /dx 2 ) = E d 2 /dx 2 = -2mE/ħ 2 ( ) Our goal is to find specific functions (x) that satisfy the equation.

41 Figure 12.13: A schematic diagram of a particle in a one-dimensional box with infinitely high potential walls

42 d 2 /dx 2 = -2mE/ħ 2 ( ) d 2 /dx 2 = (constant) I.e., consider the function Asin(kx), A, k : constants

43 Boundary conditions?

44 The boundary conditions for the particle in a box enforce the following facts : 1. The particle cannot be outside the box - it is bound inside the box. 2. In a given state the total probability of finding the particle in the box must be The wave function must be continuous.

45 Based on 1 : k =?, =Asin(kx), so that (0)=0 & (L)=0 (L)=Asin(kL)=0, k=n /L (n=1,2,3, )

46 Based on 2 : A =? = L/2

47 Quantum numbers

49 Figure 12.14: The first three energy levels for a particle in a one-dimensional box

50 n 2 h 2 E 8 ma 2 a sin n a x

53 The squared wave functions are the probability densities, and they show the difference between classical and quantum mechanical behavior. Classical mechanics : the electron has equal probability of being at any point in the box Quantum mechanics : different probabilities at different locations in the box.

54 2.2.2 Quantum Numbers and Atomic Wave Functions The particle-in-a-box example shows how a wave function operates in one dimension. Mathematically, atomic orbitals are discrete solutions of the 3-D Schrödinger equations. The same methods used for the 1-D box can be expanded to the 3-D for atoms.

55 (s, p, d ) (p x, p y, p z )

56 r =R(r)Θ( )Φ( ) = R(r) (, ) Θ( )Φ( ) : angular functions R(r) : radial functions x = r sin cos y = r sin sin z = r cos Nodal surfaces : also R(r)=0 or

58 # of nodes?

60 # of nodes?

61 r 2 R

62 Figure 12.16: (a) The probability distribution for the hydrogen 1s orbital in three-dimensional space. (b) The probability density of the electron at points. Electron density & electron probability mean the same thing. 7d 62

into successive thin spherical shells. (b) The radial probability distribution.

63 Figure 12.17: (a) Cross section of the hydrogen 1s orbital probability distribution divided into successive thin spherical shells. (b) The radial probability distribution. a 0 = Å (Bohr radius) The radial probability distribution : a plot of 4 r 2 R 2 v.s. r, where R represents the radial part of the wave function. 7d 63

64 See figure 2-7 See table 2-3 For the Molecular Orbital purpose The first derivative equals zero! Node.

65 Node :

67 Each 2p orbital has two lobes. There is a planar node normal to the axis of the orbital (so the 2p x orbital has a yz nodal plane). Each 3p orbital has four lobes. There is a planar node normal to the axis of the orbital (so the 3p x orbital has a yz nodal plane, for instance). Apart from the planar node there is also a spherical node that partitions off the small inner lobes. Each 4p orbital has six lobes. There is a planar node normal to the axis of the orbital (so the 4p x orbital has a yz nodal plane, for instance). Apart from the planar node there are also two spherical node that partition off the small inner lobes. A gallery of atomic & molecular orbitals

69 2-2-3 The Aufbau Principle 1. Electrons are placed in orbitals to give the lowest total energy to the atom. 2. Pauli exclusion principle. 3. Hund s rule of maximum multiplicity

71 c : Coulombic energy of repulsion e : exchange energy, which arises purely from quantum mechanical considerations. This energy depends on the number of possible exchanges between two electrons with the same energy & spin.

74 The Coulombic energy, c, is positive & is nearly constant for each pair of electrons. The exchange Energy, e, is negative & is also nearly constant for each possible exchange of electrons with the same spin. When the orbitals are degenerate ( 簡併 ), both favor the unpair configuration.

75 If there is a difference in energy between the levels involved, this difference, in combination with the total pairing energy, determines the final configuration (ligand-field theory, i.e. Pt 2+ -d 8 ).

77 2-2-4 Shielding In atoms with more than one electron, energies of specific levels are difficult to predict quantitatively. A useful approach to such predictions uses the concept of shielding : each electron acts as a shield for electrons farther from the nucleus, reducing the attraction between the nucleus & the more distant electrons.

80 (a) 2s 2 2p 5 : for a particular 2p electron S = 6 x 3.5 = 2.10 (b) For the 3s electron of Na : the group (2s 2, 2p 6 ) has S = 8 x 0.85 = 6.80

81 Z* = Z - S Responsible for the Octet rule & Lanthanide contraction (85%)

82 n-1 same n

83 The left same n

84 n-2 n-1 same n

86 Penetration effect : Note that although an electron in the 3s orbital spends most of its time far from the nucleus & outside the core electrons (the electrons in the 1s, 2s, & 2p orbitals), which shields it from the nuclear charge, it has a small but significant probability of being quite close to the nucleus. This effect also helps to explain why the 4s orbital fills before the 3d orbital as the order : E 4s < E 3d. (E ns < E np < E nd < E nf )

87 Figure : The radial distribution of electron probability density for the sodium atom. E 3s < E 3p < E 3d

88 Figure : Radial probability distributions

89 Note that the most probable distance of the electron from the nucleus for the 3d orbital is less than that for the 4s orbital. However, the 4s orbital allows more electron penetration close to the nucleus & thus is preferred over the 3d orbital.

90 2-3 Periodic properties of atoms Ionization energy The ionization energy, also known as the ionization potential, is the energy required to remove an e from a gaseous atom or ion : A n+ (g) A (n+1)+ (g) + e - ionization energy = U Where n=0 (first ionization energy), (second, third, ).

91 2-3-2 Electron affinity Electron affinity can be defined as the energy required to remove an e from a negative anion : A - (g) A (g) + e - electron affinity = U (or EA) [A (g) + e - A - (g) electron affinity = EA]

92 Be : 1s 2 2s 2 B : 1s 2 2s 2 2p 1 N : 1s 2 2s 2 2p 3 O : 1s 2 2s 2 2p 4

Chapter 1 - Basic Concepts: atoms

Chapter 1 - Basic Concepts: atoms Discovery of atomic structure Rutherford (1910) JJ Thomson (1897) Milliken (1909) Rutherford (1911) 1 Symbol p + e - n 0 Mass (amu) 1.0073 0.000549 1.00870 Discovery 1919,